Solve Equations 10x-2y-3z=205,-2x+10y-2z=154,-2x-y+10z=120 using Relaxation method

Solution:
Total Equations are `3`

`10x-2y-3z=205`

`-2x+10y-2z=154`

`-2x-y+10z=120`


The residuals from equations, we get
`R_1=205-10x+2y+3z`

`R_2=154+2x-10y+2z`

`R_3=120+2x+y-10z`

The table for operation is

	`R_1`	`R_2`	`R_3`
`deltax`	-10	2	2
`deltay`	2	-10	1
`deltaz`	3	2	-10

Solution steps are
`1^(st)` Approximation

`R_1=205-0=205`

`R_2=154-0=154`

`R_3=120-0=120`

Maximum is `R_1=205`

`deltax=205/10=20.5`

`2^(nd)` Approximation

`R_1=205-10*20.5=205-205=0`

`R_2=154-(-2)*20.5=154--41=195`

`R_3=120-(-2)*20.5=120--41=161`

Maximum is `R_2=195`

`deltay=195/10=19.5`

`3^(rd)` Approximation

`R_1=0-(-2)*19.5=0--39=39`

`R_2=195-10*19.5=195-195=0`

`R_3=161-(-1)*19.5=161--19.5=180.5`

Maximum is `R_3=180.5`

`deltaz=180.5/10=18.05`

`4^(th)` Approximation

`R_1=39-(-3)*18.05=39--54.15=93.15`

`R_2=0-(-2)*18.05=0--36.1=36.1`

`R_3=180.5-10*18.05=180.5-180.5=0`

Maximum is `R_1=93.15`

`deltax=93.15/10=9.315`

`5^(th)` Approximation

`R_1=93.15-10*9.315=93.15-93.15=0`

`R_2=36.1-(-2)*9.315=36.1--18.63=54.73`

`R_3=0-(-2)*9.315=0--18.63=18.63`

Maximum is `R_2=54.73`

`deltay=54.73/10=5.473`

`6^(th)` Approximation

`R_1=0-(-2)*5.473=0--10.946=10.946`

`R_2=54.73-10*5.473=54.73-54.73=0`

`R_3=18.63-(-1)*5.473=18.63--5.473=24.103`

Maximum is `R_3=24.103`

`deltaz=24.103/10=2.4103`

`7^(th)` Approximation

`R_1=10.946-(-3)*2.4103=10.946--7.2309=18.1769`

`R_2=0-(-2)*2.4103=0--4.8206=4.8206`

`R_3=24.103-10*2.4103=24.103-24.103=0`

Maximum is `R_1=18.1769`

`deltax=18.1769/10=1.8177`

`8^(th)` Approximation

`R_1=18.1769-10*1.8177=18.1769-18.1769=0`

`R_2=4.8206-(-2)*1.8177=4.8206--3.6354=8.456`

`R_3=0-(-2)*1.8177=0--3.6354=3.6354`

Maximum is `R_2=8.456`

`deltay=8.456/10=0.8456`

`9^(th)` Approximation

`R_1=0-(-2)*0.8456=0--1.6912=1.6912`

`R_2=8.456-10*0.8456=8.456-8.456=0`

`R_3=3.6354-(-1)*0.8456=3.6354--0.8456=4.481`

Maximum is `R_3=4.481`

`deltaz=4.481/10=0.4481`

`10^(th)` Approximation

`R_1=1.6912-(-3)*0.4481=1.6912--1.3443=3.0355`

`R_2=0-(-2)*0.4481=0--0.8962=0.8962`

`R_3=4.481-10*0.4481=4.481-4.481=0`

Maximum is `R_1=3.0355`

`deltax=3.0355/10=0.3035`

`11^(th)` Approximation

`R_1=3.0355-10*0.3035=3.0355-3.0355=0`

`R_2=0.8962-(-2)*0.3035=0.8962--0.6071=1.5033`

`R_3=0-(-2)*0.3035=0--0.6071=0.6071`

Maximum is `R_2=1.5033`

`deltay=1.5033/10=0.1503`

`12^(th)` Approximation

`R_1=0-(-2)*0.1503=0--0.3007=0.3007`

`R_2=1.5033-10*0.1503=1.5033-1.5033=0`

`R_3=0.6071-(-1)*0.1503=0.6071--0.1503=0.7574`

Maximum is `R_3=0.7574`

`deltaz=0.7574/10=0.0757`

`13^(th)` Approximation

`R_1=0.3007-(-3)*0.0757=0.3007--0.2272=0.5279`

`R_2=0-(-2)*0.0757=0--0.1515=0.1515`

`R_3=0.7574-10*0.0757=0.7574-0.7574=0`

Maximum is `R_1=0.5279`

`deltax=0.5279/10=0.0528`

`14^(th)` Approximation

`R_1=0.5279-10*0.0528=0.5279-0.5279=0`

`R_2=0.1515-(-2)*0.0528=0.1515--0.1056=0.2571`

`R_3=0-(-2)*0.0528=0--0.1056=0.1056`

Maximum is `R_2=0.2571`

`deltay=0.2571/10=0.0257`

`15^(th)` Approximation

`R_1=0-(-2)*0.0257=0--0.0514=0.0514`

`R_2=0.2571-10*0.0257=0.2571-0.2571=0`

`R_3=0.1056-(-1)*0.0257=0.1056--0.0257=0.1313`

Maximum is `R_3=0.1313`

`deltaz=0.1313/10=0.0131`

`16^(th)` Approximation

`R_1=0.0514-(-3)*0.0131=0.0514--0.0394=0.0908`

`R_2=0-(-2)*0.0131=0--0.0263=0.0263`

`R_3=0.1313-10*0.0131=0.1313-0.1313=0`

Maximum is `R_1=0.0908`

`deltax=0.0908/10=0.0091`

`17^(th)` Approximation

`R_1=0.0908-10*0.0091=0.0908-0.0908=0`

`R_2=0.0263-(-2)*0.0091=0.0263--0.0182=0.0444`

`R_3=0-(-2)*0.0091=0--0.0182=0.0182`

Maximum is `R_2=0.0444`

`deltay=0.0444/10=0.0044`

`18^(th)` Approximation

`R_1=0-(-2)*0.0044=0--0.0089=0.0089`

`R_2=0.0444-10*0.0044=0.0444-0.0444=0`

`R_3=0.0182-(-1)*0.0044=0.0182--0.0044=0.0226`

Maximum is `R_3=0.0226`

`deltaz=0.0226/10=0.0023`

`19^(th)` Approximation

`R_1=0.0089-(-3)*0.0023=0.0089--0.0068=0.0157`

`R_2=0-(-2)*0.0023=0--0.0045=0.0045`

`R_3=0.0226-10*0.0023=0.0226-0.0226=0`

Maximum is `R_1=0.0157`

`deltax=0.0157/10=0.0016`

`20^(th)` Approximation

`R_1=0.0157-10*0.0016=0.0157-0.0157=0`

`R_2=0.0045-(-2)*0.0016=0.0045--0.0031=0.0077`

`R_3=0-(-2)*0.0016=0--0.0031=0.0031`

Maximum is `R_2=0.0077`

`deltay=0.0077/10=0.0008`


Solution By Relaxation Method.
`x=sum deltax=31.9997~=32`

`y=sum deltay=25.9998~=26`

`z=sum deltaz=20.9995~=21`

Iterations are tabulated as below

Iteration	Operation	`deltax` (10)	`deltay` (10)	`deltaz` (10)	`R_1`	`R_2`	`R_3`
1	`x=y=z=0`	0	0	0	205	154	120
2	`deltax=205/10=20.5`	20.5	0	0	0	195	161
3	`deltay=195/10=19.5`	0	19.5	0	39	0	180.5
4	`deltaz=180.5/10=18.05`	0	0	18.05	93.15	36.1	0
5	`deltax=93.15/10=9.315`	9.315	0	0	0	54.73	18.63
6	`deltay=54.73/10=5.473`	0	5.473	0	10.946	0	24.103
7	`deltaz=24.103/10=2.4103`	0	0	2.4103	18.1769	4.8206	0
8	`deltax=18.1769/10=1.8177`	1.8177	0	0	0	8.456	3.6354
9	`deltay=8.456/10=0.8456`	0	0.8456	0	1.6912	0	4.481
10	`deltaz=4.481/10=0.4481`	0	0	0.4481	3.0355	0.8962	0
11	`deltax=3.0355/10=0.3035`	0.3035	0	0	0	1.5033	0.6071
12	`deltay=1.5033/10=0.1503`	0	0.1503	0	0.3007	0	0.7574
13	`deltaz=0.7574/10=0.0757`	0	0	0.0757	0.5279	0.1515	0
14	`deltax=0.5279/10=0.0528`	0.0528	0	0	0	0.2571	0.1056
15	`deltay=0.2571/10=0.0257`	0	0.0257	0	0.0514	0	0.1313
16	`deltaz=0.1313/10=0.0131`	0	0	0.0131	0.0908	0.0263	0
17	`deltax=0.0908/10=0.0091`	0.0091	0	0	0	0.0444	0.0182
18	`deltay=0.0444/10=0.0044`	0	0.0044	0	0.0089	0	0.0226
19	`deltaz=0.0226/10=0.0023`	0	0	0.0023	0.0157	0.0045	0
20	`deltax=0.0157/10=0.0016`	0.0016	0	0	0	0.0077	0.0031
	Total	31.9997	25.9998	20.9995

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